Theorem flimclslem 24308

Description: Lemma for flimcls 24309. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
flimcls.2	⊢ 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))

Assertion

Ref	Expression
flimclslem	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))

Proof of Theorem flimclslem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		flimcls.2	. . 3 ⊢ 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
2		topontop 23236	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3	2	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
4		eqid 2760	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	neisspw 23430	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 ∪ 𝐽)
6	3, 5	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 ∪ 𝐽)
7		toponuni 23237	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
8	7	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 = ∪ 𝐽)
9	8	pweqd 4573	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝒫 𝑋 = 𝒫 ∪ 𝐽)
10	6, 9	sseqtrrd 3945	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
11		toponmax 23249	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
12		elpw2g 5294	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
13	11, 12	syl 18	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
14	13	biimpar 483	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
15	14	3adant3 1150	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
16	15	snssd 4743	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ⊆ 𝒫 𝑋)
17	10, 16	unssd 4190	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋)
18		ssun2 4177	. . . . . 6 ⊢ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})
19	11	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 ∈ 𝐽)
20		simp2 1155	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
21	19, 20	ssexd 5285	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
22	21	snn0d 4732	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ≠ ∅)
23		ssn0 4354	. . . . . 6 ⊢ (({𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∧ {𝑆} ≠ ∅) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
24	18, 22, 23	sylancr 599	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
25	20, 8	sseqtrd 3944	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
26		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ ((cls‘𝐽)‘𝑆))
27	4	neindisj 23440	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ (𝐴 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥 ∩ 𝑆) ≠ ∅)
28	27	expr 462	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥 ∩ 𝑆) ≠ ∅))
29	3, 25, 26, 28	syl21anc 851	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥 ∩ 𝑆) ≠ ∅))
30	29	imp 412	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑥 ∩ 𝑆) ≠ ∅)
31		elsni 4597	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑆} → 𝑦 = 𝑆)
32	31	ineq2d 4117	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑆} → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝑆))
33	32	neeq1d 3014	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑆} → ((𝑥 ∩ 𝑦) ≠ ∅ ↔ (𝑥 ∩ 𝑆) ≠ ∅))
34	30, 33	syl5ibrcom 250	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ {𝑆} → (𝑥 ∩ 𝑦) ≠ ∅))
35	34	ralrimiv 3153	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → ∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅)
36	35	ralrimiva 3154	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅)
37		simp1 1154	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘𝑋))
38	4	clsss3 23382	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
39	3, 25, 38	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ ∪ 𝐽)
40	39, 26	sseldd 3909	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ ∪ 𝐽)
41	40, 8	eleqtrrd 2863	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ 𝑋)
42	41	snssd 4743	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ⊆ 𝑋)
43		snnzg 4731	. . . . . . . . . 10 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑆) → {𝐴} ≠ ∅)
44	43	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ≠ ∅)
45		neifil 24204	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
46	37, 42, 44, 45	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
47		filfbas 24172	. . . . . . . 8 ⊢ (((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
48	46, 47	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
49		ne0i 4286	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑆) → ((cls‘𝐽)‘𝑆) ≠ ∅)
50	49	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
51		cls0 23403	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
52	3, 51	syl 18	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘∅) = ∅)
53	50, 52	neeqtrrd 3029	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅))
54		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑆 = ∅ → ((cls‘𝐽)‘𝑆) = ((cls‘𝐽)‘∅))
55	54	necon3i 2987	. . . . . . . . 9 ⊢ (((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅) → 𝑆 ≠ ∅)
56	53, 55	syl 18	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ≠ ∅)
57		snfbas 24190	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅ ∧ 𝑋 ∈ 𝐽) → {𝑆} ∈ (fBas‘𝑋))
58	20, 56, 19, 57	syl3anc 1398	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ∈ (fBas‘𝑋))
59		fbunfip 24193	. . . . . . 7 ⊢ ((((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋) ∧ {𝑆} ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅))
60	48, 58, 59	syl2anc 596	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥 ∩ 𝑦) ≠ ∅))
61	36, 60	mpbird 260	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
62		fsubbas 24191	. . . . . 6 ⊢ (𝑋 ∈ 𝐽 → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
63	19, 62	syl 18	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
64	17, 24, 61, 63	mpbir3and 1361	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋))
65		fgcl 24202	. . . 4 ⊢ ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
66	64, 65	syl 18	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
67	1, 66	eqeltrid 2864	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐹 ∈ (Fil‘𝑋))
68		fvex 6898	. . . . . 6 ⊢ ((nei‘𝐽)‘{𝐴}) ∈ V
69		snex 5396	. . . . . 6 ⊢ {𝑆} ∈ V
70	68, 69	unex 7761	. . . . 5 ⊢ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V
71		ssfii 9414	. . . . 5 ⊢ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
72	70, 71	ax-mp 5	. . . 4 ⊢ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
73		ssfg 24196	. . . . . 6 ⊢ ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
74	64, 73	syl 18	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
75	74, 1	sseqtrrdi 3949	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ 𝐹)
76	72, 75	sstrid 3919	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝐹)
77		snssg 4740	. . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
78	21, 77	syl 18	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
79	18, 78	mpbiri 261	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
80	76, 79	sseldd 3909	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ 𝐹)
81	76	unssad 4191	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
82		elflim 24295	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
83	37, 67, 82	syl2anc 596	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
84	41, 81, 83	mpbir2and 726	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ (𝐽 fLim 𝐹))
85	67, 80, 84	3jca 1146	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556  {csn 4584  ∪ cuni 4866  ‘cfv 6538  (class class class)co 7420  ficfi 9405  fBascfbas 21671  filGencfg 21672  Topctop 23216  TopOnctopon 23233  clsccl 23341  neicnei 23420  Filcfil 24169   fLim cflim 24258

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1o 8479  df-2o 8480  df-en 8977  df-fin 8980  df-fi 9406  df-fbas 21680  df-fg 21681  df-top 23217  df-topon 23234  df-cld 23342  df-ntr 23343  df-cls 23344  df-nei 23421  df-fil 24170  df-flim 24263

This theorem is used by:  flimcls  24309